Created by T Madas Question 146 (***+) x x x θ x x A wire of total length 60 cm is to be cut into two pieces The first piece is bent to form an equilateral triangle of side length x cm and the second piece is bent to form a circular sector of radius x cm The circular sector subtends an angle of θ radians at the centre a) Show that xθ = 60 − x The total area of the two shapes is A cm b) Show clearly that A= ( ) − 10 x + 30 x c) Use differentiation to find the value of x for which A is stationary d) Find, correct to three significant figures, the maximum value of A , justifying the fact that it is indeed the maximum value of A SYN-F , x ≈ 7.26 , Amax ≈ 109 Created by T Madas Created by T Madas Question 147 (***+) y y = 2x − x A O x The figure above shows part of the curve with equation y = 2x − x , x ≥ The curve meets the x axis at the point A a) Show that the coordinates of A are ( 8,0 ) b) Find the exact area of the finite region bounded by the curve and the x axis area = 32 Created by T Madas Created by T Madas Question 148 (***+) non calculator y= − 3x x + , x >0 3x a) Find an expression for dy dx b) Show that the value of d2y where x = , is dx 96 dy −1 = − x −2 − x dx Created by T Madas